effect of fiber orientation on stress concentration factor in a laminate with central circular hole...

7/28/2019 Effect of Fiber Orientation on Stress Concentration Factor in a Laminate With Central Circular Hole Under Transverse Static Loading

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Indian Journal of Engineering & Materials SciencesVol. 15, December 2008, pp. 452-458

Effect of fibre orientation on stress concentration factor in a laminate with central

circular hole under transverse static loading

N D Mittal* & N K Jain

Department of Applied Mechanics, Maulana Azad National Institute of Technology, Bhopal 462 007, India

Received 12 April 2007; revised received 17 June 2008

The effect of fibre orientation (θ) on stress concentration factor (SCF) in a rectangular composite laminate with centralcircular hole under transverse static loading has been studied by using finite element method. The percent variations indeflection with fibre orientation are also compared with deflection in laminate without hole. Studies are carried out for three

D/A ratios (where D is hole diameter and A is plate width). The results are obtained for four different boundary conditions.

Three different types of materials are used for whole analysis to find the sensitivity of stress concentration with elastic

constants. A finite element study is made for whole analysis of laminate with a central hole under transverse static loading.Keywords: Finite element method, Stress concentration factor, Composite, Laminate, material properties, Fibre orientation,

Transverse loading

A laminated composite plate with central circular hole

have found widespread applications in various fields

of engineering such as aerospace, marine, automobile

and mechanical. Stress concentration arises from any

abrupt change in geometry of plate under loading. As

a result, stress distribution is not uniform throughout

the cross-section. Failures such as fatigue cracking

and plastic deformation frequently occur at points of

stress concentration. Hence, for the design of alaminated composite plate with central circular hole,

stress concentration factor plays an important role and

accurate knowledge of stresses and stress

concentration factor at the edges of hole under in

plane or transverse loading are required. Analytical

solutions are available in the literature for prediction

of SCF in different types of abrupt changes in shape.

Shastry and Raj1 have analysed the effect of fibre

orientation for a unidirectional composite laminate

with finite element method by assuming a plane stress

problem under in plane static loading. Paul and Rao2,3

presented a theory for evaluation of stressconcentration factor of thick and FRP laminated plate

with the help of Lo-Christensen-Wu higher order

bending theory under transverse loading. Xiwu

et al.4,5

evaluated stress concentration of finite

composite laminates with elliptical hole and multiple

elliptical holes based on classical laminated plate

theory. Iwaki6 worked on stress concentrations in a

plate with two unequal circular holes. Ukadgaonker

and Rao7 proposed a general solution for stresses

around holes in symmetric laminates by introducing a

general form of mapping function and an arbitrary

biaxial loading condition into the boundary

conditions. Ting et al.8

presented a theory for stress

analysis by using rhombic array of alternating method

for multiple circular holes. Chaudhuri9 worked on

stress concentration around a part through hole

weakening a laminated plate by finite elementmethod. Mahiou and Bekaou10

studied for local stress

concentration and for the prediction of tensile failure

in unidirectional composites. Toubal et al.11 studied

experimentally for stress concentration in a circular

hole in composite plate. Younis12 investigated by

reflected photoelasticity method that the assembly

stress are the result of contact and bearing stresses

between the bolts and member, contributes to

reducing stresses around the circular holes in a plate

under uniaxial tension. Peterson18 has developed good

theory and charts on the basis of mathematical

analysis and presented excellent mythology ingraphical form for evaluation of stress concentration

factors in isotropic plates with different types of

abrupt change, but no results are presented for

orthotropic and laminated plate.

In this paper, a study of rectangular laminated

composite plate with central circular hole for theeffect of fibre orientation on stress concentration

factor under transverse static loading is made. The

analytical treatment for such type of problem is very

difficult and hence the finite element method is__________*For correspondence (E-mail: [email protected])



MITTAL & JAIN: STRESS CONCENTRATION FACTOR IN A LAMINATE 453

adopted for whole analysis. The purpose of thisresearch work is to investigate the effect of fibre

orientations on SCF for normal stress in X , Y

directions (σx, σy), shear stress in XY plane (τxy) andvon mises (equivalent) stress (σeqv) in a single layer

laminate plate with central circular hole. Three typesof different composite materials of different material

properties are used for analysis to find out the

sensitivity of SCF with respect to elastic constants.

The work also illustrates the variation of SCF versus

D/A ratio in a lamina at different fibre orientations.The deflections in transverse direction (U z) for

different cases are also calculated.

Description of Problem

To study the influence of fibre orientation upon

deflection and SCF for different stresses, a laminated

composite plate of dimension 200 mm × 100 mm × 1

mm with a central circular hole of diameter D

subjected to a total transverse static load of P Newton

(which is uniformly distributed on whole plate) for all

cases is analysed by finite element method. The

analysis is carried out for three different D/A ratios.

Figure 1a shows the basic model of the problem.

Finite Element Analysis

An eight nodded linear layered structural 3-D shell

element with six degrees of freedom at each node

(specified as Shell99 in ANSYS package) was

selected based on convergence test and used through

out the study. Each node has six degrees of freedom,

making a total 48 degrees of freedom per element. In

order to construct the graphical image of thegeometries of the three different models for different

D/A ratios, a laminated plate examined using the

ANSYS (Advanced Engineering Simulation). It was

necessary to input the basic geometric elements such

as points, lines and arcs. Mapped meshing are usedfor all models so that more elements are employed

near the hole boundary. Due to the un-symmetricnature of different models investigated, it was

necessary to discretize the full laminated plate for

finite element analysis. Main task in finite elementanalysis is selection of suitable element type.

Numbers of checks and convergence test are made forselection of suitable element type from different

available elements and to decide the element length.

Results were then displayed by using post processor

of ANSYS programme. For some simple problems of

plates, the finite elements results are also assessedwith available theoretical and experimental results in

literature and it in concluded that the finite elements

results are acceptable. Figure 1b provides the example

of the discretized models for D/A =0.2, used in study.

Results and Discussion

Numerical results are presented for three different

D/A ratio as 0.1, 0.2 and 0.5. Three different

orthotropic composite materials are used for analysis.

The material properties are given in Table 1.

Where; E , G and µ represent modulus of elasticity,

modulus of rigidity and poisson’s ratio respectively.

Four types of plates (a)-(d) are analysed. In plate (a)

all edges are simply supported, in plate (b) one edge is

Fig. 1a — Details of model analysed in study (A laminated plate

with central hole under uniformly distributed static loading of P

Newton in transverse direction)

Table 1—The material properties

Materials

Properties

Boron/

aluminium

Silicon carbide/

ceramic

Woven glass/

epoxy

E x

E y

E z

Gxy

Gyz

Gzx

µxy

µyz

µzx

235 GPa

137 GPa

137 GPa

47 GPa

47 GPa

47 GPa

0.3

0.3

0.3

121 GPa

112 GPa

112 GPa

44 GPa

44 GPa

44 GPa

0.2

0.2

0.2

29.7 GPa

29.7 GPa

29.7 GPa

5.3 GPa

5.3 GPa

5.3 GPa

0.17

0.17

0.17

Fig. 1b — Typical example of finite element mesh for D/A=0.2



INDIAN J. ENG. MATER. SCI., DECEMBER 2008454

fixed, in plate (c) two edges are simply supported andtwo edges are fixed, in plate (d) all edges are fixed.

Figure 2 provides the boundary conditions at all edges

of plates (a), (b), (c) and (d).The variation of SCF for different stresses and

percent variation in U z with different fibreorientations are presented in Figs 3-11. It has been

noted that these are the maximum values in the plates.

In case of plates (a) and (c), the maximum stress

concentration for all stresses is always occurred on

boundary of hole, i.e., values of SCF for differentstresses are plotted for boundary of hole, where, in

case of plates (b) and (d), the maximum stress

concentration is occurred on supports, i.e., values of

SCF for different stresses are plotted for supports.

Maximum U z is always occurred at boundary of hole,hence, the percent variation in U z is plotted for

boundary of hole in all the cases.

Variations of SCF for σx, σy, τxy for different D/A ratios with respect to fibre orientations in plates (a),

(b), (c), and (d) made of different composite materialsare shown in Figs 3-5. Following observation can be

Fig. 2 — Boundary conditions at all edges of plates (a), (b), (c)

and (d)

Fig. 3 — Variation of SCF (for σx, σy, τxy) versus fibreorientations in plates (a), (b), (c) and (d) of boron/aluminum

material

Fig. 4 — Variation of SCF (for σx, σy, τxy) versus fibreorientations in plates (a), (b), (c) and (d) of silicon

carbide/ceramic material

Fig. 5 — Variation of SCF (for σx, σy, τxy) versus fibreorientations in plates (a), (b), (c) and (d) of woven glass/epoxy

material




made from Figs 3-5. In case of plate (a); for D/A=0.1and 0.2, maximum SCF is obtained for σx for almost

all the values of θ and attaining maximum at θ=90°,

but for D/A=0.5 maximum SCF is obtained for τxy foralmost all the values of θ and attaining maximum at

θ=90° for all materials. Figures illustrate that at anyfibre orientation, SCF for σx, σy, τxy decrease with

increase of D/A ratio for all materials. It is also

clear from figures that SCF for σx, σy, τxy obtained

maximum when θ=90° for all D/A ratios andmaterials. For all D/A ratios and materials, it has been

seen that SCF for σy is always lesser then SCF for σx

at almost all the values of θ. Maximum value of SCFis coming as 3.5 in case of woven glass/epoxy

composite material at θ=90° for D/A=0.1 for σx. Incase of plate (b); maximum SCF is obtained for τxy for

almost all the values of θ and attaining maximum

value at θ=90° for all D/A ratios and materials. For all

Fig. 6 — Variation of SCF (for σeqv) versus fibre orientations inplates (a), (b), (c) and (d) of boron/aluminum material

Fig. 7 — Variation of SCF (for σeqv) versus fibre orientations in

plates (a), (b), (c) and (d) of silicon carbide/ceramic material

Fig. 8 — Variation of SCF (for σeqv) versus fibre orientations inplates (a), (b), (c) and (d) of woven glass/epoxy material

Fig. 9 — Percent variation in U z versus fibre orientations in plates

(a), (b), (c) and (d) of boron/aluminum material




made from figures. In case of boron/aluminiummaterial, SCF increases continuously when θ changes

from 0° to 15°, decreases when θ changes from 15° to

30°, again increases when θ changes from 30° to 75°,attaining a maximum value when orientation is at 75°

and then decreases when θ changes from 75° to 90°for all D/A ratios. In case of silicon carbide/ceramic

material, SCF decreases continuously when θ changes

from 0° to 45°, attaining a minimum value when

orientation is at 45° and then again increases when θ

changes from 45° to 90°, attaining a maximum valuewhen orientation is at 90° for all D/A ratios. In case of

woven glass/epoxy, SCF increases continuously when

θ changes from 0° to 15°, decreases when θ changes

from 15° to 45°, again increases when θ changes from

45° to 75°, attaining a maximum value when

orientation is at 75° and then decreases when θ

changes from 75° to 90° for all D/A ratios. Maximum

SCF are coming as 2.2 at θ=75°, 1.9 at θ=90° and 2.1

at θ=75° for D/A=0.1 in boron/aluminium, silicon

carbide/ceramic and woven glass/epoxy composite

materials respectively. It is observed that the SCF for

σeqv also follows a symmetric trend with respect to

90° in all cases. For woven glass/epoxy laminate, SCF

follows a symmetric trend with respect to 45° when

orientation changes from 0°

to 90° and to135°

when

orientation changes from 90° to 180°. It is clear from

figures that, for all materials and D/A ratios,

maximum stress concentration occurred in case of

plate (a) for all values of θ and for plate (a) SCF

varied from 1.3 to 2.3 for different cases. It is also

observed that, in case of plate (c), some significant

stress concentration occurred. But in case of plates (b)

and (c), the effect of stress concentration is much

small, and in case of plate (d), it is almost negligible

for all cases. For plate (d), the variation of SCF with

respect to θ is also negligible for all D/A ratios and

materials; SCF is fluctuated near about 1 for all cases.

In case of plate (b); it has been seen that the effect of

D/A ratio on SCF is negligible for all values of θ and

materials. In case of all plates, the trend of variation

of SCF with respect to θ is different for different

material, i.e., variation of SCF depends up on elastic

constants. In case of plate (a); SCF obtained always

greater then 1.0 for all values of θ, D/A ratios and

materials but in case of plates (b), (c), and (d), SCF

obtained less then 1.0 in some cases.

The variation of percent variation in U z for

different D/A ratios with respect to θ in plates (a), (b),

(c), and (d) made of different composite materials are

shown in Figs 9-11. The percent variation in U Z hasbeen calculated with respect to laminate without hole

for same case. Following observation can be made

from Figs 9-11. In case of plates (a), (b) and (c), U zincreases with increase in D/A ratio, but in case of

plate (d) U z increases when D/A ratio increase from0.1 to 0.2 and then decreases when D/A ratio increase

from 0.2 to 0.5 for all values of θ and materials. For

boron/aluminium and silicon carbide/ceramic plates

(a), (c) and (d), the maximum and minimum

deflection occurred at θ=90° and θ=0° respectively,but in case of plate (b) maximum deflection occurred

when θ=0° and minimum occurred when θ=90°. In

case of woven glass/epoxy material; percent variation

in U z is almost constant with respect to θ for all D/A

ratios and plates (maximum variation is obtained up

to 5%). It has been observed that maximum percent

variation occurred for plate (a) and minimum

occurred for plate (d). It has been also seen that, per

cent variation in U z is obtained less then 0% at some

values of θ for all D/A ratios, plates and materials.

Conclusions

In general; for plates (a) and (c), the maximum

stress concentration is always occurred on hole

boundary and in case of plates (b) and (d), the

maximum stress concentration is occurred on

supports. The SCF for σx, σy, σeqv play an importantrole in plate (a), a significant role in plate (c) and

negligible role in plates (b) and (d). The SCF for τ xy

plays, an important role in plates (b), (c), (d) and a

significant role in plate (a). It has been observed that

SCF for all stresses decrease with increase in D/A

ratio, where deflection increases with increase in D/A

ratio for almost all values of θ, materials and plates.

For plates (a), (c) and (d), maximum U z always

occurred at θ=90° and for plate (b), maximum U z

always occurred at θ=0° for all D/A ratios. Maximum

SCF for τ xy always occurred at θ=90° for all cases. It

is also observed that SCF for all stresses anddeflection follow a symmetric trend with respect to

90° fibre orientation. In case of composite materials

those have same modulus of elasticity in X and Y

directions SCF for all stresses and deflection follow a

symmetric trend with respect to 45° when orientation

changes from 0 or 90° and to135° when orientation

changes from 90° or 180°. In case of all plates, the

trend of variation of SCF with respect to θ is different

for different material, i.e., variation of SCF depends

up on elastic constants. It has been also seen that the



INDIAN J. ENG. MATER. SCI., DECEMBER 2008458

SCF is most sensitive to material properties anddirectly depend on the ratio of E x / E y and E x / Gxy. The

results obtained, show that for higher values of these

ratios, SCF for all stresses may also be higher.

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effect of fiber orientation on stress concentration factor in a laminate with central circular hole...

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